NEW YORK (Reuters) - Oil prices rose slightly on Tuesday as Saudi exports fell and solid demand soaked up some of what is seen as an oversupplied market, but Ecuador's decision to opt out of an OPEC-led supply reduction pact complicated the outlook.

Saudi Arabia's crude oil exports in May fell to 6.924 million barrels per day (bpd) from 7.006 million bpd in April, official data showed on Tuesday.

The top oil exporter's goal remains to stabilize oil markets by drawing down the global inventory overhang, a Saudi industry source familiar with the kingdom's oil policy said on Tuesday.

Meanwhile in a sign of strong demand, data on Monday showed refineries in China increased crude throughput in June to the second highest on record.

But many markets are well supplied and oil for prompt delivery is trading at heavy discounts to forward futures in several parts of the world.

As a result, crude oil prices are trading at only around half the levels seen three years ago.

A deal by the Organization of the Petroleum Exporting Countries with Russia and other non-OPEC producers to cut supplies by around 1.8 million barrels per day until March 2018 has so far failed to tighten the market or push up prices.

Although many OPEC countries have restricted production, others including Nigeria and Libya are allowed to increase output.

Ecuador said it would no longer comply with an agreed OPEC production cut of 26,000 bpd due to the country's financial difficulties.

Oil Minister Carlos Perez said Ecuador was cutting only 60 percent of that figure, putting current output at 545,000 bpd.

While Ecuador is a small producer, in a note RBC Capital Markets wrote that "it could embolden other cash strapped producers to seek an exit (from the OPEC deal) as well."

"We highlight Iraq as the most important ‘at-risk’ OPEC member," the RBC note said, adding the "Iraqi oil minister ... has repeatedly criticized the terms of the November 2016 agreement, insisting that Iraq should have been exempted like Libya and Nigeria and that the (210,000 bpd) cut imposes too high a financial burden on the war-ravaged country."

U.S. oil production is also rising steadily.

The U.S. Energy Department said in a report on Monday U.S. shale oil output was likely to rise for the eighth consecutive month in August, climbing 112,000 bpd to 5.585 million bpd.

Oil prices briefly pared gains in post-settlement trade after data from the American Petroleum Institute (API) showed a surprise build of 1.6 million barrels in crude stocks for last week. Analysts polled by Reuters had forecast a draw of 3.2 million barrels. The market will watch official inventory data on Wednesday morning from the U.S. Energy Department's Energy Information Administration.

(Additional reporting by Christopher Johnson in London, Henning Gloystein in Singapore; Editing by Chris Reese, Diane Craft and David Gregorio)